where $Δx$ is the change in displacement vector and $Δt$ is the change in time

It is the velocity of the object, calculated in the shortest instant of time possible (calculated as the time interval $Δt$ tends to zero).

Mathematically, $lim_{Δt→0}ΔtΔS =dtdS $

Let's solve a problem on instantaneous velocity for better understanding

Q: A bullet fired in space is traveling in a straight line and its equation of motion is $S(t)=4t+6t_{2}$

If it travels for 15 seconds before impact, find the instantaneous velocity and acceleration at the 10th second

Solution:
We know the equation of motion:
$S(t)=4t+6t_{2}$$dtdS =dtd(4t+6t_{2}) =4+12t$

Therefore, $V_{Inst}$ at (t =10) = 4 + (12 x 10) = 124 m/s. The Instantaneous Velocity of the bullet is 124 m/s

Instantaneous Acceleration is defined as: $lim_{Δt→0}ΔTΔV $, where $ΔV$ is the change in Velocity vector

Mathematically, $lim_{Δt→0}ΔTΔV =dTdV =d_{2}Td_{2}S $ In the problem, $d_{2}Td_{2}S =12$

Hence, Instantaneous Acceleration of the bullet is $12m/s_{2}$

Revision

Average Speed is defined as the ratio of the total distance covered to the total time taken. It does not indicate constant velocity of travel

Velocity of a particle/object is a vector with a specified direction. It's magnitude is the speed of the particle/object. Hence, Velocity is nothing but speed with a direction

Instantaneous quantities are defined at an instant of time. Examples: Instantaneous Velocity of an object is defined as it's Velocity at an instant of time